The references people provided are pretty good. But maybe my description of the basics will also help.
The goal of the game is to destroy the enemy citadel before he destroys yours. You will start near your citadel, and protecting your citadel and your side of the map are defensive towers. In order to get to the enemy citadel, you'll have to destroy some of his towers.
You also have portals, which spawn groups of warriors (aka creeps). These warriors can be defeated easily by a demigod, but they will generally just walk down a path and meet the enemy team's warriors. With the help of a demigod, these warriors may defeat the enemy warriors and reach a tower where they'll immediately be killed.
However, once the demigods build up a supply of money, they can upgrade these waves so that they contain more than just warriors. With enough upgrades, these waves of creeps will eventually get catapults and giants, and begin to do serious damage to the enemy's towers, and eventually the citadel.
Demigods build up money over time by controlling gold mines. In every map, there will be gold mines in your base that you'll always control, and there may or may not be gold mines out in the middle of the map that you can take control of. Demigods can also get money by killing enemy demigods.
You can use this money to buy things: you can buy items from the item shop in your base, you can buy citadel upgrades (such as the upgrades for your creep waves) at your citadel, and you can buy artifacts at the artifact shop (usually in the middle of the map but sometimes in your base). Items will improve your demigods stats, such as health, armor, damage, etc. Some can be used to heal yourself or damage your enemies, like a healing potion. Artifacts are more powerful items that are mostly too expensive to purcahse until mid-game.
Throughout the map, there are also flags. These flags can be controlled by one of the teams, and you will start out controlling the flags near your base, while the enemy controls the flags near his base. There will also be flags in the middle of the map that are neutral. If you control a flag, you will get bonuses, which you can see by mousing-over the flag. You can take control of a flag for your team by standing on it for a minute, and when you capture it, it will change to your color. Of course, the enemy can later capture it back by doing the same thing.
As you kill creeps, capture flags, and kill enemy demigods, your demigod will gain levels. At each level, you get a skill point that you can assign to one of your skills. You can do this at any time, and usually you should do it as soon as you gain a level.
At any point, if your demigod dies, you will respawn at your base after about 35 seconds.
When starting the game, usually you'll first visit the item shop in your base and use your starting money to buy some items. I often see people purchase boots of speed to start, or sometimes I'll see people purchase a healing potion and some armor, or sometimes people will just purchase a lot of potions. After you purchase your items, you should place your first skill point.
At level 1, with only the starting items, you will be too weak to defeat a tower, so your primary goal will be to gain a few levels. The best way to do this is by capturing flags and killing enemy creeps. You can also try to kill enemy demigods, but it may be difficult to do, since following them into their towers will result in your death early on. Killing enemy demigods is good for gaining money, but it won't get you as much experience as creeps and flags.
As you gain more levels and get better items, you'll be better able to take out enemy towers and kill enemy demigods. But you'll want to pay attention to the enemy demigod's level, since especially as a new player, they may be leveling much faster than you. Or they could be lower than you and be an easy target. You can see an enemy demigod's level by mousing-over it and looking at its health bar in the bottom right.
The goal is to become more powerful than the enemy demigods and/or outmaneuver them to destroy their towers and citadel.
Also, your team will gain "war rank" over time. This war rank increases faster if you control more flags. You need to have a certain war rank in order to purchase higher-level citadel upgrades. You can see your team's war rank by pressing tab.
A few tips:
The creeps are colored differently for each team, so you can tell whether they are yours by looking at their color. But sometimes its difficult to see the color. In this case, you can mouse-over them, and if they highlight green, they are freindly. If they highlight red, they are enemies. You can also look at the direction the creeps are moving. If they are moving away from your base, then they are probably friendly. If they're moving toward your base, then they're probably hostile.
Remember to purcahse the spawn wave citadel upgrades when you get a high enough war rank. These are the upgrades that give you priests, then angels, then catapultasauri, then giants. These will be very helpful in destroying the enemy base. There may be some games where you don't need these upgrades, but if you aren't experienced enough to recognize that situation, then it's best to err on the side of purchasing them: they can really change the tide of battle.
When you are choosing your skills, you will see the level requirement for the skill at the top of the skill selection screen. Similarly, the war rank required for a citadel upgrade is displayed at the top of the citadel upgrade purchasing screen.
Don't underestimate the auto-attack damage of your demigod, especially assassins and Oak. With some good items and an artifact, these generals can attack very fast and do a lot damage with each hit.
General demigods can buy special items called idols at the item shop that let them summon more units using the white circles to the right of your mana bar. There are three types of idols: minotaurs, archers/gunners, and priests. You can have one idol of each type.
The assassin demigods are Torchbearer, Unclean Beast, Regulus, and Rook, and they're generally easier to play than the generals. (The generals are Oak, Queen of Thorns, Sedna, and Lord Erebus). Assassins tend to do a lot of damage, while generals can summon minions to help them.
As you play games, you will earn favor points (you get more if you win, and even more if you get awards). You can use these points to purchase favor items, which you choose at any time by clicking the fourth item slot.
Hopefully this helped.